An elliptic Garnier system

Abstract

We present a linear system of difference equations whose entries are expressed in terms of theta functions. This linear system is singular at $4m+12$ points for $m\ge 1$, which appear in pairs due to a symmetry condition. We parameterize this linear system in terms a set of kernels at the singular points. We regard the system of discrete isomonodromic deformations as an elliptic analogue of the Garnier system. We identify the special case in which $m=1$ with the elliptic Painlevé equation, hence, this work provides an explicit form and Lax pair for the elliptic Painlevé equation.

27 pages, 2 figures

References in corpus (5)

• A Lax Formalism for the Elliptic Difference Painlevé Equation
• Commutation Relations and Discrete Garnier Systems
• Padé interpolation for elliptic Painlevé equation
• The Lattice Structure of Connection Preserving Deformations for q-Painlevé Equations I
• The noncommutative geometry of elliptic difference equations

Cited by in corpus (5)

• Variations of $q$-Garnier system
• An Elliptic Garnier System from Interpolation
• A Variation of the $q$-Painlevé System with Affine Weyl Group Symmetry of Type $E_7^{(1)}$
• D-type Minimal Conformal Matter: Quantum Curves, Elliptic Garnier Systems, and the 5d Descendants
• Connection Problem for an Extension of $q$-Hypergeometric Systems